Metric, stratifiable and uniform spaces of \(G\)-permutation degree (Q6550098)

Let \(X=(X,\tau)\) be a (topological) space and \(\tau_c:=\{X\backslash U:U\in\tau\}\) the collection of closed subsets of \(X\). An \textbf{open stratification} (resp., \textbf{closed stratification}) of \(X\) is a subset sequence assignment \(\tau\rightarrow\tau^\mathbb{N}\), \(U\mapsto \{U_n\}_{n=1}^\infty\) (resp., \(\tau\rightarrow\tau_c{}^\mathbb{N}\), \(U\mapsto \{U_n\}_{n=1}^\infty\)) such that for all \(U,V\in\tau\), (i) each closure \(\overline{U}_n\subset U\), (ii) \(U=\bigcup_{n=1}^\infty U_n\), and (iii) if \(V\subset U\) then each \(V_n\subset U_n\). A space is \textbf{metrizable} (resp., \textbf{uniformizable}) if it is homeomorphic to a metric space (resp., uniform space), \textbf{stratifiable} if it is \(T_1\) and has an open stratification [\textit{C. J. R. Borges}, Pac. J. Math. 17, 1--16 (1966; Zbl 0175.19802)], \textbf{semi-stratifiable} if it has a closed stratification [\textit{G. D. Creede}, ibid. 32, 47--54 (1970; Zbl 0189.23304)], and \textbf{semi-metrizable} if it is both stratifiable and first countable [\textit{T. G. Raghavan} and \textit{I. L. Reilly}, Indian J. Pure Appl. Math. 18, 219--225 (1987; Zbl 0622.54022)].\N\NConsider a space \(X\), a positive integer \(n\geq 1\), the symmetric/permutation group \(S_n\) of \(\{1,2,\dots,n\}\), and a subgroup \(G\) of \(S_n\). We know that \(G\) acts on \(X^n=\{x=(x_1,\dots,x_n):x_i\in X\}\) by the map \(G\times X^n\rightarrow X^n\), \((\sigma,x)\mapsto x_\sigma:=(x_{\sigma(1)},\dots,x_{\sigma(n)})\). Let \([x]_G:=\{x_\sigma:\sigma\in G\}\) denote the \textbf{\(G\)-orbit} of \(x\) in \(X^n\). The \textbf{\(G\)-permutation degree} of \(X\) is the set of orbits SP\(^n_GX:=\{[x]_G:x\in X^n\}\) viewed as the quotient space of \(X^n\) with quotient map \(\pi^n_G:X^n\rightarrow\textrm{SP}^n_GX\), \(x\mapsto[x]_G\). The operation SP\(^n_G\) defines a covariant functor on (specific) categories of spaces according to [\textit{V. V. Fedorchuk}, Russ. Math. Surv. 36, No. 3, 211--233 (1981; Zbl 0495.54008); translation from Usp. Mat. Nauk 36, No. 3(219), 177--195 (1981)] and the problem ``If property \(P\) holds for a space \(X\), is the same true about SP\(^n_GX\)?'' is of great interest.\N\NThis paper studies metrizability (via Theorem 3.3), semi-metrizability (via Theorem 4.6), stratifiability (via Theorem 4.2), semi-stratifiability (via Theorem 4.4), and uniformizability (via Theorem 5.3) of the space SP\(^n_GX\), proving among other things that, for each of these properties, SP\(^n_GX\) has the property if and only if \(X\) has the property (with \(X\) assumed to be \(T_1\) in the stratifiability and semi-stratifiability results). With regard to the \textbf{universality problem} (i.e., whether or not, in a given class of spaces, there exists a \textbf{universal element}, as one into which all other elements of the class can be embedded) from [\textit{S. D. Iliadis}, Universal spaces and mappings. Amsterdam: Elsevier (2005; Zbl 1072.54001)], certain classes of uniformizable \(T_0\) spaces are noted to have universal elements, while some open problems are posed concerning the existence of universal elements in the classes of metrizable, sem-metrizable, stratifiable, and semi-stratifiable spaces.